For an analysis of the new schemes, we have extended convergence results from Dai and Liao [1]} in Section . In view of the TBB steps, instead of \(\lambda _1 \le \alpha _k \le \lambda _n\) , we have studied the more general setting \(\xi _{\rm {low}}\, \lambda _1 \le \alpha _k \le \xi ^{\rm {up}}\, \lambda _n\) , particularly for \(\frac{1}{2} < \xi _{\rm {low}}\le 1\)  and \(\xi ^{\rm {up}}\ge 1\) .
